Czechoslovak Mathematical Journal, Vol. 67, No. 2, pp. 427-437, 2017

A new characterization of symmetric group by NSE

Azam Babai, Zeinab Akhlaghi

Received December 26, 2015. First published March 20, 2017.

Abstract: Let $G$ be a group and $\omega(G)$ be the set of element orders of $G$. Let $k\in\omega(G)$ and $m_k(G)$ be the number of elements of order $k$ in $G$. Let nse$(G) = \{m_k(G) k \in\omega(G)\}$. Assume $r$ is a prime number and let $G$ be a group such that nse$(G)=$ nse$(S_r)$, where $S_r$ is the symmetric group of degree $r$. In this paper we prove that $G\cong S_r$, if $r$ divides the order of $G$ and $r^2$ does not divide it. To get the conclusion we make use of some well-known results on the prime graphs of finite simple groups and their components.

Keywords: set of the numbers of elements of the same order; prime graph